Мини-курс МЛ ТИ: Владимир Александрович Гурвич «Булева двойственность и разрешимость по Нэшу игр двух лиц»

Polynomial_algorithms_computing_two_lexicographica 

In 1975 the first author proved that every finite tight two-person game form g is Nash-solvable, that is, for every payoffs u and w of two players the obtained normal form game (g; u, w) has a Nash equilibrium (NE) in pure strategies. Several proofs of this theorem were obtained later. Here we strengthen the result and give a new proof, which is shorter than previous ones. We show that game g; u, w) has two types of NE, realized by a lexicographically safe (lexsafe) strategy of one player and some special best response of the other. The proof is constructive, we obtain a polynomial algorithm computing these lexsafe NE. This is trivial when game form g is given explicitly. Yet, in applications g is
frequently realized by an oracle O such that size of g is exponential in the size |O| of O. We assume that game form g = g(O) generated by O is tight and that an arbitrary ±1 game (g; u0, w0) (in which payoffs u0 and w0 are zero-sum and take only values ±1) can be solved in time polynomial in |O|. These assumptions allow us to compute two (one for each player) lexsafe NE in time polynomial in |O|. These NE may coincide. We consider four types of oracles known in the literature and show that all four satisfy the above assumptions.
Keywords: Nash equilibrium, Nash-solvability, game form, tightness, deterministic graphical game structure, game in normal and in positional form, monotone bargaining, veto voting, Jordan game.
AMS subjects: 91A05, 94D10, 06E30